Quadratic systems with two invariant real straight lines and an invariant ellipse

Abstract After the linear differential systems the easiest nonlinear differential systems are the quadratic polynomial differential systems or simply the quadratic systems. These systems have been studied intensively and there are more than one thousand papers published on them. Here we study all quadratic systems possessing two real invariant straight lines (taking into account their multiplicities) and one invariant ellipse. By analysing the relative positions of the finite and infinite equilibria and of their separatrices with respect to these invariant curves, we determine the four topologically distinct phase portraits of this class of quadratic systems in the Poincaré disc. These four phase portraits already have appeared before in the classification of the phase portraits of the quadratic systems having an invariant ellipse, and three of these phase portraits also appeared before in the classification of the phase portraits of the quadratic systems having the infinity in the Poincaré compactification filled with equilibria, but in these previous classifications it was unknown that such phase portraits can have exactly two real invariant straight lines (taking into account their multiplicities) and one invariant ellipse.